Wythoff Array
المؤلف:
Kimberling, C.
المصدر:
"Fractal Sequences and Interspersions." Ars Combin. 45
الجزء والصفحة:
...
11-1-2021
1135
Wythoff Array
The Wythoff array is an interspersion array that can be constructed by beginning with the Fibonacci numbers ![<span style=]()
{F_2,F_3,F_4,F_5,...}" src="https://mathworld.wolfram.com/images/equations/WythoffArray/Inline1.gif" style="height:15px; width:116px" /> in the first row and then building up subsequent rows by iteratively adding ![<span style=]()
{F_(3+k),F_(4+k),F_(5+k),F_(6+k),...}" src="https://mathworld.wolfram.com/images/equations/WythoffArray/Inline2.gif" style="height:15px; width:160px" />, where 
or 1 is the smallest offset producing an initial term that has not occurred in a previous row. This process gives the array
 |
(1)
|
Read by skew diagonals from lower left to upper right, this gives the sequence 1; 4, 2; 6, 7, 3; ... (OEIS A083412), while read by skew diagonals from upper right to lower left, this gives 1; 2, 4; 3, 7, 6; ... (OEIS A035513).
The first column is given by 1, 4, 6, 9, 12, 14, 17, ... (OEIS A003622), with the initial term of the 
th row given by
with 
is the golden ratio. Rows numbered 
, i.e., 2, 5, 7, 10, 13, ... (OEIS A001950) have offset 
, while rows numbers 
, i.e., 1, 3, 4, 6, 8, ... (OEIS A000201) have 
.
The element 
can be given explicitly by
 |
(4)
|
REFERENCES:
Fraenkel, A.; and Kimberling, C. "Generalized Wythoff Arrays, Shuffles and Interspersions." Disc. Math. 126, 137-149, 1994.
Kimberling, C. "Stolarsky Interspersions." Ars Combin. 39, 129-138, 1995.
Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157-168, 1997.
Kimberling, C. "Interspersions and Dispersions." https://faculty.evansville.edu/ck6/integer/intersp.html.
Sloane, N. J. A. "My Favorite Integer Sequences." In Sequences and Their Applications (Proceedings of SETA '98) (Ed. C. Ding, T. Helleseth, and H. Niederreiter). London: Springer-Verlag, pp. 103-130, 1999. https://www.research.att.com/~njas/doc/sg.pdf.
Sloane, N. J. A. "The Wythoff Array and the Para-Fibonacci Sequence." https://www.research.att.com/~njas/sequences/classic.html.
Sloane, N. J. A. Sequences A000201/M2322, A001950/M1332, A003622/M3278, and A083412 in "The On-Line Encyclopedia of Integer Sequences."
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